1
www.PosterPresentations.com Acceleration of the Moon as a point mass w.r.t. the Earth • Point mass interactions • Non-relativistic • 3372 normal points, June 1996 – August 2012 (data sources: CDDIS and APOLLO website) • Mean: 5.7 cm, Standard deviation: 4.8 cm (with 11 % data rejected) Abstract Components of the Model Abstract 1. Graduate University for Advanced Studies (SOKENDAI), 2. Hitotsubashi University, 3. National Astronomical Observatory of Japan Ryosuke Nagasawa 1 , Toshimichi Otsubo 2 , Hideo Hanada 1,3 Development of software for high-precision LLR data analysis laser path observatory reflector In order to determine the lunar orbital/rotational motion and tidal deformation using lunar laser ranging (LLR) observation data, analysis software is being developed. As the first step of the study, we construct an LLR observation model, combining the newest physical models. The observation model consists of the lunar orbit and libration obtained from DE430 (provided by NASA JPL), and the other physical models compatible with IERS Conventions (2010) such as Earth orientation, solid Earth/Moon tides, and some factors affecting propagation delay. For the purpose of calculating these components precisely, we use the modules of the geodetic data analysis software "c5++" (Otsubo et al., JpGU, 2011). LLR observation data are provided as normal points obtained at Apache Point, Grasse, Matera and McDonald. In this calculation, there are 3372 normal points distributed from June 1996 to August 2012. Comparing the observed and calculated one-way range, the mean and the standard deviation of the residuals are about 5.7 cm and 4.8 cm respectively. 3 steps of this study Current status Step 1: LLR observation modeling Almost finished (this report) Step 2: Lunar orbit/rotation integration Unfinished Step 3: Lunar orbit/rotation determination Not yet started Next Step: Orbit Integration (preliminary) AP11 AP14 AP15 LU17 LU21 Apache Point 255 264 668 75 81 Grasse 135 126 418 0 25 Matera 1 1 10 0 0 McDonald 164 165 977 0 7 Table 1. Number of normal points we compared Lunar orbit around the Earth, Librations • JPL lunar and planetary ephemeris DE430 Station coordinates and velocities • ITRF 2008 + Station eccentricities (～1m) • Apache Point Obs. Coordinates and velocities (by Prof. Müller, personal communication, 2013) Solid Earth tide • IERS Conv. (2010) model Earth rotation • IERS Conv. (2010) models + EOP 08 C04 data • ERA, precession, nutation, polar motion, UT1 – UTC correction Reflector coordinates • Williams et al. (2013) Solid Moon tide • Murphy et al. (2009) • Love numbers from Williams et al. (2013) Aberration of laser light • Fukushima (ed.) (2009) • Solving the equation of light time iteratively, using Newton method Tropospheric propagation delay • Mendes et al. (2002) model Relativistic propagation delay • IERS Conv. (2010) Chapter. 11 • Effects of perturbers on one-way range: Sun ≈ 7 - 8 [m], Earth ≈ 30 - 40 [cm], Moon ～ 0.1 [mm] TDB – TT (time scale transformation) • Kovalevsky et al. (1989) • Needed to compensate for the difference in time scales between ephemeris and observed round-trip time • Ephemeris DE430: Barycentric Dynamical Time (TDB) • Round-trip time: Terrestrial Time (TT) • TT = UTC + 32.184 sec + accumulated Leap Seconds ✓ Ryosuke Nagasawa [email protected] 18 th International Workshop on Laser Ranging, 11-15 Nov., 2013. 2 − 1 − 12 = 1+ 2 ln 1 + 2 + 12 1 + 2 − 12 =1 ℎ 2 = 0.0476, 2 = 0.0107 Δ = 0 3 ℎ 2 3 2 ⋅ 2 − 1 2 + 3 2 ⋅ − ⋅ Acknowledgement Figure 1. Lunar tidal displacements in mean Earth direction Lorentz transformation (spatial components) • IERS Conv. (2010) Chapter. 11 • Spatial transformation accompanied by time transformation TDB – TT Results models calculated by using modules of “c5++” c5++ = 1− 2 − − 1 2 2 ⋅ Figure 2. Illustration of aberration of light t 1 : known t 2 t 3 ・Station coordinates ・Earth rotation Earth rotation angle Precession UT1 - UTC Nutation Polar motion ・Solid Earth tides ～ 100 m ～ 100 m ～ 10 m ～ 10 m ～ 0.1 m ・Aberration of light ・Tropospheric delay ・Relativistic propagation delay ・4-dim. space-time transformation (Lorentz transformation b/w TDB and TT) ～ 1 m ～ 1 m ～ 1 m ～ 1 m orders of magnitudes on one-way range ・Lunar orbit around the Earth ・Libration ・Solid Moon tide ～ 0.1 m 13-Po14 Figure 5. Residuals (Number of single raw ranges) Figure 6. Residuals (Number of photons) The software “c5++” is developed in the collaboration among three Japanese research groups which are located at Hitotsubashi University, NICT, and JAXA. We would like to thank Prof. Jürgen Müller, Leibniz Universität Hannover, for providing the ITRF coordinate and estimated velocity for the Apache Point laser ranging station. Figure 7. Comparison b/w two versions of ephemeris DE421 (2008) and DE430 (2013) Figure 8,9. Absolute values of lunar acceleration w.r.t. Earth, contributed by solar system bodies (unit: [m/s 2 ], Calculated by using DE430) Figure 10. Residuals of lunar orbit around the Earth b/w DE430 and tentative orbit integrated by using c5++ (x, y, z: ICRF coordinates) • Components: Sun, Solar system planets, Earth J 2 (very preliminary study) • Cowell method, double-precision calculation • Initial condition: fixed to DE430 • Remaining factors (≲1 cm) → ocean tide loading, atmospheric loading, and target signatures • Normal points observed before 1996 need to be considered • Relativistic (Einstein-Infeld-Hoffmann equation): ≲ 10e-10 m/s 2 • Figure effect of the Earth and the Sun : ≲ 10e-09 m/s 2 … mostly Earth J 2 • Earth tides effect : The tides raised upon the Earth affect the motion of the Moon ● ● ● ● ● Figure 3,4. One-way residuals (observed - calculated) by stations and reflectors
